Browning butter is a sure way to suffuse a dish with a great deal of nutty, buttery flavor without using a lot of fat. Example: Sautéed Chicken with Sage Browned Butter. But the process is a little tricky because once the butter begins to brown, it can race right into burnt. Then nutty becomes bitter.
Success depends on visual cues, so use a stainless steel pan—you can see the butter change color better. Use no more than medium heat so that the browning proceeds gradually. First the butter will foam in the pan: The milk solids are separating from the butterfat, and the water is evaporating. Then the foam subsides and the milk solids begin to brown. Now the butter gives off its characteristic nutty aroma (the French call brown butter beurre noisette, or hazelnut butter). Some recipes call for adding lemon juice at this point; the tartness complements the sweet butter, while the juice cools it and slows the browning. Either way, when the butter turns amber-brown, take the pan off the heat. If you're not using it immediately (say, drizzling it over steamed vegetables), get it out of the hot pan and into a bowl so the residual heat doesn't continue to push the butter from brown to burnt.